Correlatd electrons: the case of high T c cuprates Introduction: Hubbard U - Mott transition, The cuprates: Band structure and phase diagram NMR as a local magnetic probe Magnetic susceptibilities NMR as a local magnetic probe The single spin fluid ptroblem Hyperfine fields Comparison between different nuclei The pesudogap Questions about the phase diagram
Beyond the independent electrons aoproximation LDA: Local interactions between electrons are replaced by an interaction with a potential associated with a mean value of the electronic density. Only valid if the electron kinetic energy is much larger than the interaction between electrons Not good if the electrons are localized in the vicinity of the atomic nucleus LDA narrow band = small kinetic energy
Mott Insulators Beyond the LDA Antiferro-magnetic t Hubbard model U Mott Insulating ground state Non doped Cuprates Most simplified model Poorly handled by LDA: Metallic ground state Excited states are not described (transitions between atomic levels) Realistic extension of the LDA: DMFT Atomic state hybridized with a medium with an effective energy spectrum
Electronic correlations: Mott -Hubbard metal insulator transition E 0 +U E 0 Set of hydrogen atoms forming a lattice with paramer d One electron per site: atomic level E 0 Coulomb repulsion U forbids double occupancy: excited level E 0 +U d If d decreases U Isolated atoms t INSULATOR W METAL the transfer integral t increases and the band width W increases Insulator to metal transition occurs for U=W U>W gives an insulating magnetic state and not a metallic state for a half filled band
SUPERCONDUCTORS Metals and alloys T c Oxides Organic
SUPERCONDUCTING CUPRATES La 2 - x Sr x CuO 4 YBa 2 Cu 3 O 6+x
PARENT CUPRATES j j < 0 b< < 0 Cu(1) 3d 10 S=0 j < 0 Cu(2) 3d 9 S=1/2
Phase Diagram and Band Structure E 3d 10 U 3d 9 O 2p band n(e) E 3d 10 3d 9 O 2p band n(e) Localized Cu holes and Oxygen band carriers? NO!! Strong O2p - Cu 3d hybridization Magnetic correlations
Magnetic susceptibilities SQUID Measures χ m ( T ) = = Ion cores orbital Spin χ χ dia dia + + [ ] orb s χ χ ( T ) + + i χ orb χ NMR shift measures on each nuclear site i s + always ( χ imp ) =(x,y,z) i = atomic sites K ( T ) = K dia i + K orb + K s ( T ) = K dia i + A orb χ + orb A s χ s ( T ) Local magnetic measurement on each nuclear site i
H.A, T. Ohno and P. Mendels, PRL 1989 89 Y NMR shift CuO 2 plane CuO 2 plane K i dia orb orb s s, ( T ) = K i + Ai, χ + Ai, χ ( T ) Local magnetic measurement But transferred hyperfine couplings
Sign if 89 Y NMR shift Negative sign comes from Y4d orbitals: core polarization pπ Y4d pσ Y4d Y4d Y4d Very weak OK So negative sign comes from pσ -Y4d hybridization
Is there an independent oxygen band at the Fermi level H.A, T. Ohno and P. Mendels, PRL 1989 K s (ppm) K s (T) = A(4d-2pσ) χ s (T) Slope = A(4d-2pσ) does not change with hole doping So there is no independent oxygen spin degree of freedom at E F
Phase Diagram and Band Structure E E 3d 10 3d 10 U 3d 9 O2pσ n(e) O2pσ n(e) 3d 9 WRONG PICTURE Magnetic correlations
Zhang-Rice singlet Spin singlets Cu3d O2pσ The holes steal spins from the copper hole background There is a single spin fluid
K M. Takigawa et al 1991, 1993 i Single spin fluid behaviour dia orb orb s s, ( T ) = K i + Ai, χ + Ai, χ ( T ) 89 Y versus 17 O 63 Cu versus 17 O A single T dependence for K a (T): due to χ Cu (T) Notice: this allows determinations of the shift references for all nuclei
What about the origin of this T dependence? H.A, T. Ohno and P. Mendels, PRL 1989 Origin for K s Large decrease (nearly full loss) of χ s (T) above T c Pseudogap in the electronic excitations
H.Alloul, T. Ohno and P. Mendels, PRL 1989 Pseudogap in the cuprates
89 Y NMR shift 0.95 T m Phase diagram and pseudogap 0.91 T m Low T decrease of the susceptibility: 0.64 T m opening of the pseudogap 89 K( T ) = σ + A χ( T ) Chemical shift Hyperfine coupling
Questions About the Phase Diagram Pseudogap: Phase transition? Crossover? Order parameter? Preformed pairs? M I SG MIT and Disorder? Strange metal? FL Transition to a Fermi liquid? Pseudogap joins Tc curve or QCP??
Examples of Magnetic Correlations Magnetic correlations: ZHANG-RICE SINGLET Magnetic correlations and ordered charge segregation: STATIC STRIPES
Generic Phase Diagram of the Cuprates? SG This shape of phase diagram is apparently generic However the optimal T c is not generic and the hole concentration is not always well determined